<trans-title xml:lang="en">Complex system reconstruction</trans-title>

Hai-Feng Zhang; Wen-Xu Wang

doi:10.7498/aps.69.20200001

Acta Physica Sinica, Volume. 69, Issue 8, 088906-1(2020)

Complex system reconstruction

Hai-Feng Zhang¹ and Wen-Xu Wang^2、*

Author Affiliations

¹School of Mathematical Science, Anhui University, Hefei 230601, China

²State Key Laboratory of Cognitive Neuroscience and Learning IDG/McGovern Institute for Brain & Research, School of Systems Science, Beijing Normal University, Beijing 100875, China

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Equations(48)

${{AX}} = {{Y}}.$

(1)

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$\begin{array}{l} \quad \min {\left\| {{X}} \right\|_0} \\ {\rm{s}}.{\rm{t}}.~~ {{AX}} = {{Y}}. \\ \end{array} $

(2)

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$\begin{array}{l} \quad \min {\left\| {{X}} \right\|_1} \\ {\rm{s}}.{\rm{t}}.~~ {{AX}} = {{Y}}. \\ \end{array} $

(3)

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${{\dot {{x}}}_i} = {{{f}}_i}\left( {{{{x}}_i}} \right) + \sum\limits_{j = 1,~j \ne i}^N {{{{C}}_{ij}} \cdot \left( {{{{x}}_j} - {{{x}}_i}} \right)} ,$

(4)

${{{C}}_{{{i}}j}} = \left[ {\begin{array}{*{20}{c}} {C_{ij}^{1,1}}&{C_{ij}^{1,2}}& \cdots &{C_{ij}^{1,D}} \\ {C_{ij}^{2,1}}&{C_{ij}^{2,2}}& \cdots &{C_{ij}^{2,D}} \\ \cdots & \cdots & \cdots & \cdots \\ {C_{ij}^{D,1}}&{C_{ij}^{D,2}}& \cdots &{C_{ij}^{D,D}} \end{array}} \right],$

(5)

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${{{\varGamma }}_i}\left( {{{{x}}_i}} \right) = {{{f}}_i}\left( {{{{x}}_i}} \right) - \sum\limits_{j = 1,~j \ne i}^N {{{{C}}_{ij}} \cdot {{{x}}_i}} .$

(6)

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$\begin{split}{\left[ {{{{\varGamma }}_i}\left( {{{{x}}_i}} \right)} \right]_k} =\;& \sum\limits_{{l_1} = 0}^n \sum\limits_{{l_2} = 0}^n \cdots \sum\limits_{{l_D} = 0}^n {{\left[ {{{\left( {{\alpha _i}} \right)}_k}} \right]}_{{l_1}, \cdots {l_D}}}\\ & \times {{\left[ {{{\left( {{{{x}}_i}} \right)}_1}} \right]}^{{l_1}}}{{\left[ {{{\left( {{{{x}}_i}} \right)}_2}} \right]}^{{l_2}}} \cdots {{\left[ {{{\left( {{{{x}}_i}} \right)}_D}} \right]}^{{l_D}}} ,\end{split}$

(7)

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$\begin{split} &{{\dot {{x}}}_i}\left( {{t_m}} \right) = {{{\varGamma }}_i}\left( {{{{x}}_i}\left( {{t_m}} \right)} \right) + \sum\limits_{j = 1,~j \ne i}^N {{{{C}}_{ij}} \cdot {{{x}}_j}\left( {{t_m}} \right)} ,\\ & \qquad \quad \left( {m = 1,2, \cdots ,M} \right),\end{split}$

(8)

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${{AX}} = {{Y}},$

(9)

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${P_i} = \sum\limits_{j = 1,~j \ne i}^N {{a_{ij}}{{S}}_i^{\rm{T}} \cdot {{P}}} \cdot {{{S}}_j}.$

(10)

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$\begin{split} &{P_i}\left( {{t_m}} \right) = \sum\limits_{j = 1,~j \ne i}^N {{a_{ij}}{{S}}_i^{\rm{T}}\left( {{t_m}} \right) \cdot {{P}}} \cdot {{{S}}_j}\left( {{t_m}} \right),\\ & \qquad \quad\left( {m = 1,2, \cdots ,M} \right),\end{split}$

(11)

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${{{G}}_i}{{{A}}_i} = {{{P}}_i}.$

(12)

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$P_i^{01}\left( t \right) = 1 - {\left( {1 - {\lambda _i}} \right)^{\sum\limits_{j = 1,j \ne i}{{a_{ij}}S_t^j} }}, $

(13)

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$\ln \left[ {1 - P_i^{01}\left( t \right)} \right] = {\rm{ln}}\left( {1 - {\lambda _i}} \right)\sum\limits_{j = 1,j \ne i}^N {{a_{ij}}S_t^j} .$

(14)

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${{{X}}_{{i}}}{{{A}}_i} = {{{Y}}_{{i}}},$

(15)

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$\begin{split} {{{A}}_i} ={}& \big[\ln \left( {1 - {\lambda _i}} \right)a_{i1}, \cdots, \ln \left( {1 - {\lambda _i}} \right){a_{i, i - 1}}, \\ {}& \ln \left( {1 - {\lambda _i}} \right){a_{i, i + 1}}, \cdots,\ln \left( {1 - {\lambda _i}} \right){a_{iN}} \big]^{\rm{T}} \end{split} $

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$P_i^{01}\left( t \right) \approx {c_i}\sum\limits_{j = 1,~j \ne i}^N {{a_{ij}}S_t^j} + {d_i}.$

(16)

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${{\dot {{x}}}_i} = {{{f}}_i}\left( {{{{x}}_i}} \right) + \sum\limits_{j = 1}^N {{J_{ij}}{{{g}}_{{{i}}j}}\left( {{{{x}}_i},{{{x}}_j}} \right)} + {{{I}}_i}\left( t \right) + {{{\eta }}_i}\left( t \right),$

(17)

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${{\dot {{x}}}_i} = {{{f}}_i}\left( {{{{x}}_i}} \right) + \sum\limits_{j = 1}^N {{J_{ij}}{{{g}}_{ij}}\left( {{{{x}}_i},{{{x}}_j}} \right)} .$

(18)

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$\begin{split} & \dot x_i^{\left( d \right)}\left( {{t_m}} \right) = f_i^{\left( d \right)}\left( {{{{x}}_{{i}}}\left( {{t_m}} \right)} \right) \\ & \qquad\qquad + \sum\limits_{j = 1}^N {{J_{ij}}g_{ij}^{\left( d \right)}\left( {{x_i}\left( {{t_m}} \right),{{{x}}_{{j}}}\left( {{t_m}} \right)} \right)}. \end{split} $

(19)

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$\dot x_{i,m}^{\left( d \right)} = f_{i,m}^{\left( d \right)} + \sum\limits_{j = 1}^N {{J_{ij}}g_{ij,m}^{\left( d \right)}}, $

(20)

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${{{J}}_{{i}}}{{{X}}_i} = {{{Y}}_i}, $

(21)

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${\dot x_i} = {f_i}\left( {{x_i}} \right) + \sum\limits_{j = 1}^N {{J_{ij}}{g_j}\left( {{x_j}} \right)} ,$

(22)

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${\dot y_i} = {f_i}\left( {{y_i}} \right) + \sum\limits_{j = 1}^N {{K_{ij}}{g_j}\left( {{y_j}} \right)} + {I_i},$

(23)

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${e_i} = {y_i} - {x_i},$

(24)

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${\dot K_{ij}} = - {\gamma _{ij}}{g_j}\left( {{y_j}} \right){e_i},$

(25)

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${\dot x_i} = \sum\limits_{j = 1}^N {{{\tilde J}_{ij}}{x_j}} + {I_i}\left( t \right),$

(26)

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$\sum\limits_{j = 1}^N {{{\tilde J}_{ij}}x_{j,m}^ * } = - {I_{i,m}}.$

(27)

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${{\tilde{{J}}}_{{i}}}{{{X}}_i} = {{{Y}}_i},$

(28)

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${\dot \phi _i} = {\omega _i} + \sum\limits_{j = 1}^N {{J_{ij}}{g_{ij}}\left( {{\phi _j} - {\phi _i}} \right)} + {I_{i,m}},$

(29)

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${\varDelta _{ij,m}} = {\phi _{i,m}} - {\phi _{j,m}}.$

(30)

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${\varOmega _m} = {\omega _i} + \sum\limits_{j = 1}^N {{J_{ij}}{g_{ij}}\left( {{\phi _{j,m}} - {\phi _{i,m}}} \right)} + {I_{i,m}}.$

(31)

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$\begin{split}{D_{i,m}}\; & = {\varOmega _m} - {\Omega _0} - {I_{i,m}} \\ &= \sum\limits_{j = 1}^N {{J_{ij}}\left[ {{g_{ij}}\left( {{\varDelta _{ij,m}}} \right) - {g_{ij}}\left( {{\varDelta _{ij,0}}} \right)} \right]} ,\end{split}$

(32)

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${\dot x_i} = {f_i}\left( {{x_i}} \right) + \sum\limits_{j = 1}^N {{J_{ij}}{g_{ij}}\left( {{x_i},{x_j}} \right)} + {I_i},$

(33)

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${I_i} = - \theta \left( {{x_i} - {{\hat x}_i}} \right).$

(34)

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$\theta \left( {{x_{i,s}} - {{\hat x}_i}} \right) = {f_i}\left( {{x_{i,s}}} \right) + \sum\limits_{j = 1}^N {{J_{ij}}{g_{ij}}\left( {{x_{i,s}},{x_{j,s}}} \right)} .$

(35)

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$\begin{split}{\varDelta _i} =\; & {f_i}\left( {{{\hat x}_i}} \right) + \sum\limits_{j = 1}^N {{J_{ij}}{g_{ij}}\left( {{{\hat x}_i},{{\hat x}_j}} \right)} \\ &- \left[ {{f_i}\left( {{x_{i,s}}} \right) + \sum\limits_{j = 1}^N {{J_{ij}}{g_{ij}}\left( {{x_{i,s}},{x_{j,s}}} \right)} } \right],\end{split}$

(36)

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$\theta \left( {{x_{is}} - {{\hat x}_i}} \right) = {f_i}\left( {{{\hat x}_i}} \right) + \sum\limits_{j = 1}^N {{J_{ij}}{g_{ij}}\left( {{{\hat x}_i},{{\hat x}_j}} \right)} - {\varDelta _i}.$

(37)

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$\theta {x_{is}} = {f_i}\left( 0 \right) + \sum\limits_{j \ne k}^N {{J_{ij}}{g_{ij}}\left( {0,0} \right) + {J_{ik}}{g_{ik}}\left( {0,{{\hat x}_k}} \right)} - {\varDelta _{i,k}}.$

(38)

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$\begin{split}\theta \left( {{x_{is,1}} - {x_{is,2}}} \right) =\; & {J_{ik}}\left[ {{g_{ik}}\left( {0,{{\hat x}_{k,1}}} \right) - {g_{ik}}\left( {0,{{\hat x}_{k,2}}} \right)} \right] \\ &- \left[ {{\varDelta _{ik,1}}- {\varDelta _{ik,2}}} \right],\\[-12pt]\end{split}$

(39)

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$\theta {s_{ik}} = \left\{\!\!\!{\begin{array}{*{20}{l}} {{J_{ik}}{\eta _{ik}} + {\lambda _{ik}},}&{{\rm{if}}}&{{a_{ik}} = 1,}\\ {{\lambda _{ik}},}&{{\rm{if}}}&{{a_{ik}} = 0.} \end{array}} \right.$

(40)

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${{\dot {{x}}}_i} = {{{f}}_i}\left( {{{{x}}_i}} \right) + c\sum\limits_{j = 1}^N {{L_{ij}}{{H}}\left( {{{{x}}_j}} \right)} + {\eta _i}\left( t \right),$

(41)

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${\dot {{\xi }}} = \left[ {D{\hat{{ F}}}\left( {{\bar {{x}}}} \right) - c{\hat {{L}}} \otimes D{\hat {{H}}}\left( {{\bar {{x}}}} \right)} \right]{{\xi }} + {{\eta }},$

(42)

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$0 = \left\langle \dfrac{{\rm{d}}({{{\xi}} {{{\xi}} ^{\rm{T}}}} )}{{\rm{d}}t} \right\rangle = - {\hat {{A}}}{\hat {{C}}} -{\hat {{C}}}{{\hat {{A}}}^{\rm{T}}} + \left\langle {{{\eta}}{{{\xi}} ^{\rm{T}}}} \right\rangle + \left\langle {{{\xi}} {{{\eta}} ^{\rm{T}}}} \right\rangle ,$

(43)

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${\hat {{A}}}{\hat {{C}}} + {\hat {{C}}}{{\hat {{A}}}^{\rm{T}}} = {\hat {{D}}},$

(44)

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$\frac1{{{k_i}}} {{\sum\limits_{j = 1,\,j \ne i}^N {{a_{ij}}S_t^j} }} \approx \frac1{{N - 1}} {{\sum\limits_{j = 1,\,j \ne i}^N {S_t^j} }},$

(45)

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$\begin{split} &P\left( {S_{t + 1}^j = 1,i \to j\left| {S_t^i = 1,S_t^j = 0} \right.} \right)\\ =\; & P\left( {i \to j\left| {S_t^i = 1,S_t^j = 0,S_{t + 1}^j = 1} \right.} \right)\\ & \times P\left( {S_{t + 1}^j = 1\left| {S_t^i = 1,S_t^j = 0} \right.} \right), \end{split}$

(46)